Problem: Simplify and expand the following expression: $ \dfrac{4y}{y + 4}+\dfrac{y}{y + 5} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(y + 4)(y + 5)$ Multiply the first term by $\dfrac{y + 5}{y + 5}$ $ \begin{align*} \dfrac{4y}{y + 4} \times \dfrac{y + 5}{y + 5} & = \dfrac{(4y)(y + 5)}{(y + 4)(y + 5)} \\ & = \dfrac{4y^2 + 20y}{(y + 4)(y + 5)}\end{align*} $ Multiply the second term by $\dfrac{y + 4}{y + 4}$ $ \begin{align*} \dfrac{y}{y + 5} \times \dfrac{y + 4}{y + 4} & = \dfrac{(y)(y + 4)}{(y + 5)(y + 4)} \\ & = \dfrac{y^2 + 4y}{(y + 5)(y + 4)}\end{align*} $ Now we have: $ = \dfrac{4y^2 + 20y}{(y + 4)(y + 5)} + \dfrac{y^2 + 4y}{(y + 5)(y + 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{4y^2 + 20y + y^2 + 4y}{(y + 4)(y + 5)} $ $ = \dfrac{5y^2 + 24y}{(y + 4)(y + 5)}$ Expand the denominator: $ = \dfrac{5y^2 + 24y}{y^2 + 9y + 20}$